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Module 52.1: Bond Valuation and Yield to Maturity
N = number of years
PMT = the annual coupon payment
I/Y = the annual discount rate
FV = the par value or selling price at the end of an assumed holding period
Take note of a couple of points here. The discount rate is entered as a whole number in
percent, 10, not 0.10. The 10 coupon payments of $100 each are taken care of in the N = 10
and PMT = 100 entries. The principal repayment is in the FV = 1,000 entry. Lastly, note that
the PV is negative; it will be the opposite sign to the sign of PMT and FV. The calculator is
just “thinking” that to receive the payments and future value (to own the bond), you must
pay the present value of the bond today (you must buy the bond). That’s why the PV amount
is negative; it is a cash outflow to a bond buyer.
Now let’s value that same bond with a discount rate of 8%:
The calculator solution is:
N = 10; PMT = 100; FV = 1,000; I/Y = 8; CPT → PV= –1,134.20
If the market discount rate for this bond were 8%, it would sell at a premium of $134.20
above its par value. When bond yields decrease, the present value of a bond’s
payments, its market value, increases.
If we discount the bond’s cash flows at 12%, the present value of the bond is:
The calculator solution is:
N = 10; PMT = 100; FV = 1,000; I/Y = 12; CPT → PV= –887
If the market discount rate for this bond were 12%, it would sell at a discount of $113 to
its par value. When bond yields increase, the present value of a bond’s payments, its
market value, decreases.
It’s worth noting here that a 2% decrease in yield-to-maturity increases the bond’s value by
more than a 2% increase in yield decreases the bond’s value. This illustrates that the bond’s
price-yield relationship is convex, as we will explain in more detail in a later topic review.
Calculating the value of a bond with semiannual coupon payments. Let’s calculate
the value of the same bond with semiannual payments.
Rather than $100 per year, the security will pay $50 every six months. With an annual
YTM of 8%, we need to discount the coupon payments at 4% per period which results
in a present value of:
The calculator solution is:
N = 20; PMT= 50; FV= 1,000; I/Y = 4; CPT → PV= –1,135.90
The value of a zero-coupon bond is simply the present value of the maturity payment.
With a discount rate of 3% per period, a 5-period zero-coupon bond with a par value of
$1,000 has a value of:
LOS 52.b: Identify the relationships among a bond’s price, coupon rate, maturity,
and market discount rate (yield-to-maturity).
CFA® Program Curriculum: Volume 5, page 407
So far we have used a bond’s cash flows and an assumed discount rate to calculate the
value of the bond. We can also calculate the market discount rate given a bond’s price
in the market, because there is an inverse relationship between price and yield. For a 3year, 8% annual coupon bond that is priced at 90.393, the market discount rate is:
N = 3; PMT = 8; FV = 100; PV = –90.393; CPT → I/Y = 12%
We can summarize the relationships between price and yield as follows:
1. At a point in time, a decrease (increase) in a bond’s YTM will increase (decrease)
2. If a bond’s coupon rate is greater than its YTM, its price will be at a premium to
par value. If a bond’s coupon rate is less than its YTM, its price will be at a
discount to par value.
3. The percentage decrease in value when the YTM increases by a given amount is
smaller than the increase in value when the YTM decreases by the same amount
(the price-yield relationship is convex).
4. Other things equal, the price of a bond with a lower coupon rate is more sensitive
to a change in yield than is the price of a bond with a higher coupon rate.
5. Other things equal, the price of a bond with a longer maturity is more sensitive to
a change in yield than is the price of a bond with a shorter maturity.
Figure 52.1 illustrates the convex relationship between a bond’s price and its yield-tomaturity:
Figure 52.1: Market Yield vs. Bond Value for an 8% Coupon Bond
Relationship Between Price and Maturity
Prior to maturity, a bond can be selling at a significant discount or premium to par
value. However, regardless of its required yield, the price will converge to par value as
maturity approaches. Consider a bond with $1,000 par value and a 3-year life paying
6% semiannual coupons. The bond values corresponding to required yields of 3%, 6%,
and 12% as the bond approaches maturity are presented in Figure 52.2.
Figure 52.2: Bond Values and the Passage of Time
Time to Maturity (in years)
YTM = 3%
YTM = 6%
YTM = 12%
The change in value associated with the passage of time for the three bonds represented
in Figure 52.2 is presented graphically in Figure 52.3. This convergence to par value at
maturity is known as the constant-yield price trajectory because it shows how the
bond’s price would change as time passes if its yield-to-maturity remained constant.
Figure 52.3: Premium, Par, and Discount Bonds
MODULE QUIZ 52.1
To best evaluate your performance, enter your quiz answers online.
1. A 20-year, 10% annual-pay bond has a par value of $1,000. What is the price of
the bond if it has a yield-to-maturity of 15%?
2. An analyst observes a 5-year, 10% semiannual-pay bond. The face amount is
£1,000. The analyst believes that the yield-to-maturity on a semiannual bond
basis should be 15%. Based on this yield estimate, the price of this bond would
3. An analyst observes a 20-year, 8% option-free bond with semiannual coupons.
The required yield-to-maturity on a semiannual bond basis was 8%, but suddenly
it decreased to 7.25%. As a result, the price of this bond:
C. stayed the same.
4. A $1,000, 5%, 20-year annual-pay bond has a YTM of 6.5%. If the YTM remains
unchanged, how much will the bond value increase over the next three years?
MODULE 52.2: SPOT RATES AND ACCRUED
LOS 52.c: Define spot rates and calculate the price of a bond using
this content is
CFA® Program Curriculum: Volume 5, page 411
The yield-to-maturity is calculated as if the discount rate for every bond cash flow is the
same. In reality, discount rates depend on the time period in which the bond payment
will be made. Spot rates are the market discount rates for a single payment to be
received in the future. The discount rates for zero-coupon bonds are spot rates and we
sometimes refer to spot rates as zero-coupon rates or simply zero rates.
In order to price a bond with spot rates, we sum the present values of the bond’s
payments, each discounted at the spot rate for the number of periods before it will be
paid. The general equation for calculating a bond’s value using spot rates (Si) is:
EXAMPLE: Valuing a bond using spot rates
Given the following spot rates, calculate the value of a 3-year, 5% annual-coupon bond.
This price, calculated using spot rates, is sometimes called the no-arbitrage price of a
bond because if a bond is priced differently there will be a profit opportunity from
arbitrage among bonds.
Because the bond value is slightly greater than its par value, we know its YTM is
slightly less than its coupon rate of 5%. Using the price of 1,001.80, we can calculate
the YTM for this bond as:
N = 3; PMT = 50; FV = 1,000; PV = –1,001.80; CPT → I/Y =4.93%
LOS 52.d: Describe and calculate the flat price, accrued interest, and the full price
of a bond.
CFA® Program Curriculum: Volume 5, page 413
The coupon bond values we have calculated so far are calculated on the date a coupon is
paid, as the present value of the remaining coupons. For most bond trades, the
settlement date, which is when cash is exchanged for the bond, will fall between coupon
payment dates. As time passes (and future coupon payment dates get closer), the value
of the bond will increase.
The value of a bond between coupon dates can be calculated, using its current YTM, as
the value of the bond on its last coupon date (PV) times (1 + YTM / # of coupon periods
per year)t/T, where t is the number of days since the last coupon payment, and T is the
number of days in the coupon period. For a given settlement date, this value is referred
to as the full price of the bond.
Let’s work an example for a specific bond: